Results

The mass of the pseudoscalar meson

We use the interpolating fields Eq. (3.3) to extract the mass of the pseudoscalar meson. These lead to the pseudoscalar-pseudoscalar two-point function GP P. In the region of light quarks, this particle is identified with the pion. Unfortunately, this correlator has systematic problems for light quark masses. We get not only

contributions from the two quarks traveling directly from the source to the sink.

There are additional contributions from quarks going around the spatial periodic boundary conditions in a topologically non-trivial way. As a rule of thumb, on a finite lattice with spatial extendL/a and a pion mass amPS, the product LmPS

should not be smaller than 4. This is the reason why we haven’t done simulations for all combinations ofβ1 and the lattice size; the physical size of the particle has to fit inside the box, too.

Furthermore, we get contributions from the chiral logarithms that are believed to arise in quenched QCD. Contrary to the linear behavior of m²_PS as a function ofmq predicted by the Gell-Mann–Oakes–Renner relation Eq. (1.31), there is an additional logarithmic term in the quenched theory [Mor87, Sha90, BG92]

m²_PS = 2Amq There have been several attempts in the last years to determine the parameter δ [JLQCD97, QCDSF00, BDET00, DDH⁺02b, DDH⁺02a, CH02, CP-PACS02, Hau02, D⁺02b]. The results scatter in the range between 0.05 and 0.48. The first results with Wilson or Kogut-Susskind quarks were in the region of 0.1, whereas the results with quarks of better chirality led to higher values in the region of 0.2. We present our result for δ below.

A third source of trouble are finite volume effects due to zero modes of the Dirac operator. They don’t contribute in full QCD due to the vanishing fermion determinant. However, the contribution of these modes can be removed [B⁺00, GHR01, D⁺02a] from the pseudoscalar-pseudoscalar two-point function GP P (with Γ =γ5) by subtracting the scalar-scalar two-point functionGSS with Γ =1 in Eq. (3.18)

GP P−SS =GP P −GSS . (3.32) The effect of this is easily understood in the spectral decomposition of the prop-agator in which Eq. (3.18) becomes for Γ =γ5

with ψ_λ the eigenvector of the Dirac operator with eigenvalue iλ. One uses the fact that the zero modes are eigenvectors ofγ5 with eigenvalue ± 1. For λ⁰ = 0 its contribution to Eq. (3.33) can be rewritten as

D X

λ

ψ¯_λ⁺(t)γ5ψλ⁰(t) ¯ψ⁺_λ0(0)γ5ψλ(0) (−iλ+m)(iλ⁰+m)

E , (3.34)

which in turn is equal to its contribution to the scalar-scalar correlation function (Γ =1)

0 5 10 15 20 25 30

Figure 3.4: The two-point function for the pion (Γ = γ5, full dots) and the subtracted G_{P P−SS} (open dots) of Eq. (3.32) for three different values of the quark mass parameter am0. It is taken from the β1 = 8.70 , 16³ ×32 sample.

The solid lines are the result of the cosh fit.

In this way, the effects of the zero modes can be removed at the cost of an additional heavy particle visible in the short time behavior of the propagator.

However, as we use an approximate solution to the Ginsparg-Wilson equation, the eigenvalues of γ5 are not ±1 but rather ≈ ±0.85. The cancelation is not perfect. We attempt the subtraction and see an improvement in the chiral limit.

We remark that the subtraction is possible only for small quark masses. For heavy quarks, the mass of the scalar and the pseudoscalar particle get closer and the separation of the two becomes very difficult.

As an example for the propagators, we plot in Fig. 3.4 the GP P and GP P−SS

two-point functions on a 16³×32 lattice withβ1 = 8.70. The quark mass increases from left to right. The effect of the of the subtraction is largest for the smallest quark mass. As the scalar particle is heavier, we see a larger deviation from the cosh fit for the subtracted two-point function than for the unsubtracted one.

The square of the extracted masses for β1 = 8.70 and β1 = 7.90 is shown in Fig. 3.5. We observe that for β1 = 8.70 the P P −SS two-point function really extrapolates linearly to 0 whereas P P bends up. Because the lattice volume is larger for β1 = 7.90 than for 8.70 (about a factor of 2 for each side), the effect subtraction is less pronounced.

Let us finally remark that we can simulate at values ofmPS/mV down to 0.3 without facing the problem of exceptional configurations (mV is the mass of the vector meson). This means that the spectrum of the Dirac operator is well enough

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 and β1 = 7.90 (right). For the smaller volume (left) the P P correlator bends up for r0m0 →0. The line is a result for the fit to a mixed data set which consists of the theP P −SS mass as long as it is smaller and r0m <0.4. Its intercept is compatible with 0 within the errorbar.

ordered at the origin such that we do not get zero or negative eigenvalues. For the Wilson operator there is a significant amount of exceptional configurations even atmPS/mV= 0.4.

The quenched chiral logarithm

The coefficient δ of the quenched chiral logarithm in Eq. (3.31) is of special interest. Chiral perturbation theory predicts a value of≈0.2 and a confirmation from the lattice would strengthen the believe in this theory. We want to extractδ from the dependence of the mass of the pseudo-scalar meson on the quark mass.

However, on the lattice there are several sources of systematic errors in particular in the region of small quark masses. We have to use a proper definition of, both, the quark mass and the pion mass such that they vanish in the same point as predicted by Eq. (3.31).

As we only have an approximate solution of the Ginsparg-Wilson equation, we have an additive renormalization of the bare quark mass. Therefore, we use the bare axial-vector Ward identity (AWI) quark massm^AWI in the following. It is defined by

2m^AWI = hP⁻(0)∂tA⁺_t(p= 0, t)i

hP⁻(0)P⁺(p= 0, t)i (3.36) withP the pseudo-scalar density and At the time component of the axial-vector current. The + and the − give the relevant flavor combinations, e.g., P^± = P^u±iP^d. The plateaus intare very flat for this ratio and the resulting errorbars

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Figure 3.6: The mass of the pseudoscalar meson divided by quark massm^AWI as a function of m^AWI for β1 = 7.90 on the 16³ ×32 lattice. The left plot is mPS

from theP P correlator, the right plot is form theP P−SS correlator alone. The solid curve correspond to fits to the data plotted. For the P P data Eq. (3.31) was fitted to the full set, for the P P −SS data we fitted to the smallest quark masses withB ≡0.

are small. It is, thus, a very convenient definition of the quark mass. However, for the physical quantity, this quark mass has to be renormalized. But we can redefine the constants A and B in Eq. (3.31) such, that the renormalization constants cancel.

We already discussed that the chiral limit of themPSis spoiled by the effect of the zero modes. In order to remove them, one can either try to mix theP P with the P P −SS data or stick to the smallest quark masses and use the P P −SS data alone. However, one has to keep in mind, that this is not the only finite size effect in the region of light quarks and, thus, we should treat differences due to the subtraction as an estimate for possible other systematic errors.

The results of the two procedures are shown in Fig. 3.6. For the P P data we fitted Eq. (3.31) to the data and plotted the curve together with the individual contributions (δ = 0.10, A = 3.1, B = 2.0), see Fig. 3.6. We observe that the large logarithm is to a large extend compensated by the quadratic term. The logarithmic and the quadratic term seem unnaturally big in the regions where they are expected to be relevant (for small and large masses, respectively). The situation is different for the P P −SS data set. There we fitted in the range m^AWI<0.06 and obtainedδ ≈0.03. However, the curve describes the data quite well for larger quark masses. The discrepancy between the two results should be taken as a warning. The finite size effect changes the result drastically, even though the subtraction of the zero modes is not perfect as the eigenvalue of γ5

is not ±1. Other finite size effects, e.g., from the size of the wave function or non-trivial winding of the quarks around the torus are not considered, yet.

Another method to extract the δ parameter is to look at the mass of the pseudoscalar meson mPS with quarks of different masses m1 and m2. Quenched chiral perturbation theory predicts for this mass

m²_PS,12 =A(m1+m2)n

In order to remove the constants A, B and Λχ the following cross ratio y was proposed [CP-PACS02]:

For small δ, α_X and small quark masses this is expected to behave like

y= 1 +δx+αXz+O(m², δ²) (3.39) as long as the quark masses are not too small such that the logarithms in the square brackets in Eq. (3.37) are not large. As z is of higher order in the quark mass, the leading behavior of y should be linear inx with slope δ. For m1 =m2

we get x= 0. Large ratios ofm1/m2 lead to values of x further away from zero.

It remains to decide, which pseudoscalar mass to use. Even though the phys-ical volume is large, the subtraction makes a difference (O(5%)) for the small quark masses. We follow the two preceding investigations and use theP P masses.

The resulting plot is shown in Fig. 3.7. We find a linear behavior and extract δ ≈ 0.18(3), which is compatible with Hauswirth’s extraction [Hau02]. Further questions arise: How can we judge the quality of the extraction this kind of plot?

What is the effect of finite size effects? In which range should we use Eq. (3.37)?

To clarify these questions, we have plotted m²_PS/(m^AWI₁ +m^AWI₂ ) as a function of (m^AWI₁ +m^AWI₂ ) for the complete non-diagonal data set atβ1 = 7.9 in Fig. 3.8.

The first observation is that up to (m^AWI₁ +m^AWI₂ ) ≈ 0.1 all data points lie on one curve; the deviations for larger quark masses are below one standard

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 -0 0.8

0.85 0.9 0.95 1 1.05

=0.16 δ

=0.21 δ y

x

Figure 3.7: The xy-plot for β1 = 7.90 on the 16³ ×32 lattice. The points are connected with solid lines for the smaller quark mass fixed; the longer this line the smaller is this fixed mass. The dotted lines correspond to 1 +δx for two values of δ.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9

+m2

m1

) 2+m 1/(m2 PSm

Figure 3.8: The mass of the pseudoscalar meson divided by the sum of the valence quark masses as a function of this sum forβ1 = 7.90 on the 16³×32 lattice. The values lie on a common line up tom1+m2 ≈0.15.

0 0.5 1 1.5 2 2.5 lattices. a is set by the Sommer parameter r0. The dotted line corresponds to a fit to the β₁ = 7.90 data.

deviation. Thus it seems questionable that there is additional information in the non-diagonal data compared to the mass degenerate case. Furthermore, the conclusion for the diagonal case, i.e. that theδ is mainly a finite size effect, holds for the non-diagonal mesons too; even though the lines in thexy-plot are straight.

All these values forδaren’t meant to be the final answer. We have listed them to visualize the large systematic uncertainties in the extraction forδ. We see that the effect of the subtraction of the zero modes is large even in our largest volume.

Hence, it seems not possible to control the systematic errors, in particular those from the effect of the zero modes and other finite size effects.

The mass of the vector meson

The lightest meson, which can be compared with the particle data book, is the vector meson that is identified with theρ in the limit of light quarks. It is gener-ated by Γ =γi with i= 1, 2, 3. We extract the mass of each of the components on jackknife subensembles, then average over the three components and calculate the jackknife error for it. The result is given in Fig. 3.9, where the scale is set by the Sommer parameterr0 = 0.5 fm calculated from the interpolating function Eq. (1.27). The square of the vector mesons mass is plotted as a function ofm²_PS for the three values of β₁ on the 16³ ×32 lattices. Each data set shows little deviation from a straight line and the three sets coincide sufficiently well. A fit-ted linear function through all points of theβ1 = 7.90 sample extrapolates these values to 830 MeV. This is 8% larger than the value in the particle data book of 770 MeV. However one has to keep in mind the various uncertainties. On top

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1

1.1 1.2 1.3 1.4 1.5 1.6 1.7

× 32 a=0.08 fm; 163

× 24 a=0.10 fm; 123

× 32 a=0.10 fm; 163

× 32 a=0.15 fm; 163

exp

mV

mN

2

/ mV 2

mPS

Figure 3.10: The APE plot for the three values ofβ1. The experimental value is plotted as a burst, the heavy quark limit is given by the star.

of the statistical error we use the quenched approximation without a continuum extrapolation and there is certainly an error in the value ofr0 = 0.5 fm. This in mind, the result is sufficiently accurate.

The proton and its parity partner

The dependence of the nucleon massmN on the pseudoscalar meson mass is often presented in a so-called APE plot. There, the ratio of the nucleon mass to the vector meson mass is plotted as a function of (mPS/mV)². In this way the lattice constant cancels and it is easy to compare the results for differentβ1. If the same ratio is plotted against (mPS/mV) it is called an Edinburgh plot.

We show the APE plot for the nucleon mass mN in Fig. 3.10. The ratio mN/mV is given for the three different values ofβ1 on the large lattices together with the results on 12²×24 atβ = 8.35. The experimental value is marked with the burst at (mπ/mρ)² = 0.033 and mp/mρ = 1.22. The other known point is in the heavy quark limit, where the masses of the particles are given entirely by the quark masses. The mesons have two and the baryons three valence quarks which leads to mPS/mV = 1 and mN/mV = 1.5. The three values of β1 suffer from different problems, in particular at small pion masses. The small β1 has a = 0.15 fm. Thus there might be larger discretization effects; we will turn to this question in Sec. 3.3.3. On the other hand, the ensemble with β1 = 8.70

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1

1.1 1.2 1.3 1.4 1.5 1.6 1.7

1.8 _{=8.70; 16}³_{× 32}

β1

× 32

=8.35; 163

β1

× 32

=7.90; 163

β1

exp

mN N*

m

2

mV 2

mPS

Figure 3.11: The ratio of the N^∗ mass to the proton mass as a function of (mPS/mV)². The experimental value of 1.62 is given by the burst. The data has been extracted on the larger lattices for all values of β1.

has a relatively small lattice. The particle hardly fits into the box and the wave function is heavily distorted. Smaller volume leads to larger mN/mV. This can be seen by comparing the 12³×32 and the 16³×32 lattices at β1 = 8.35.

It is interesting to look at the mass of the parity partner of the proton, the N^∗. Whereas the nucleon mass is given by the exponential decay for t ¿ T, the mass of the N^∗ is given by the slope in the region T > t À 0, see Fig. 3.3.

We have extracted the N^∗ mass on the large lattices and plotted its ratio to the nucleon mass in Fig. 3.11. The result if given down to (mPS/mV)² = 0.3 as the errors became too large for smaller values of this ratio. Nevertheless, it is encouraging as the finer lattices show a linear behavior that extrapolates well to the experimental point.

Dispersion relations

The relativistic energy momentum relation in the continuum reads with the speed of lightc= 1, the energyE and the rest mass of the particle m

m² =E² −p² . (3.42)

A priori, it isn’t clear that this is valid on the lattice too. However, the more continuum properties are realized at finite lattice spacing, the easier is the contin-uum limit. The various Dirac operators perform very differently in this respect,

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Figure 3.12: The dispersion relation for theπon the left and theρon the right for three different values of the quark mass parameter. The lines are the continuum relation Eq. (3.42) starting atp = 0. The data is from the 16³×32 sample with β₁ = 7.90.

see [GH00] for a comprehensive analysis in two dimensions. Whereas the Perfect Action is constructed such that Eq. (3.42) is fulfilled for all momenta in the Bril-louin zone, the overlap operator starting from the Wilson action shows a very poor result. We now look at the dispersion relations for the nucleon, the pseu-doscalar and the vector meson for the chirally improved operator. On a lattice with lattice sizeL the momentum can take discrete values

pµ= 2π

We use all combinations of momenta with|p| ≤4π/L. The energies are averaged over the different directions where possible.

The dispersion relations for the pseudoscalar and the vector meson are shown for β1 = 7.90 and three different values of the quark mass parameter am0 in Fig. 3.12. Within the errorbars we have a good agreement with the continuum relations indicated by the straight lines.

Another measure for the quality of the dispersion relation is the following quantity, which should be 1 for perfect agreement with the continuum result and is sometimes called speed of light c

c= E²−m²

p² . (3.44)

It is plotted for three bare quark masses and momentum |ap| = 2π/L for β1 = 7.90 on the 16³ ×32 lattices in Fig. 3.13. There is no statistically significant deviation from the continuum dispersion relation.

0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22

Figure 3.13: The speed of light for the nucleon, pseudoscalar and vector meson.

The results are slightly displaced. The parameters are β1 = 7.90 and 8.35 on 16³×32 with |ap|= 2π/L.

Im Dokument Chiral symmetry and hadronic measurements on the lattice (Seite 69-80)